I have a few questions...
Can I look at my opponent's Mana Zone? If yes, can I know which card is removed from the Mana Zone to somewhere else by a card effect?

If a card is moved to the Shield Zone by a card effect, do the shields need to be shuffled or the new shield is placed with public knowledge?

Please reply as detailed as possible, so that it will make things easier. Thank you!

Both mana zones are face up so that first question basically answers itself... as for the shield part, no the shield zone doesnt have to be shuffled, there's no rules stating that that must occur when cards are placed into the shield zone